SLOCUM. — FINITE CONTINUOUS GROUPS, 2i9 



r r r 



(A) x! = Xi+ 2fc ofj X, Xi + i 2, :S, a^ Oi A;. X; x, + ... = (i (x, a) 

 1 11 



(i = 1, 2 . . . r) ; 

 further, that, if aud only if 



r 



will this family satisfy differential equations of the form required by the 

 first fundamental theorem. Consequently, only if this criterion is satis- 

 fied by the infinitesimal transformations can they generate a group. 



Proceeding now, as in the. demonstration of the first fundamental 

 theorem, we introduce certain new parameters [x, and, finally, obtain the 

 equation 



where a^ = 0^ (fj., a) (^-=1,2,... r). As before, since the family of 

 transformations 9Ea, defined now by equations (A), contains the identical 

 transformation, we have 



where a^ =: </>* (jx, a""), and fij. = ^j. (a, a'"') (k = 1, 2, . . . r)* and thus 



Ea Ea = Ea- 



In the former case we saw, page 247, that, if the a's were chosen arbitra- 

 rily, one or more of the fx's might be infinite. In the present case the 

 /x's are numerical multiples of the a's f ; aud, consequently, the /a's are 

 finite whenever the a's are finite. E.g. (n = r = 2), 



fi (x, a) — Xi -\- tto, 



12 {x, «) == e«2 Xo -f- Oi 



02«'"= (e°2 — M2 — 1) /^2^^- (^"^~ ''= — 1) 



/^l («2 — /^2) 



/A2(e°2-M2 — 1)' 

 a, ^ f/'2 (ilAj «) = «2 /^2J 



which give ai = ^/Ji (//, a*"') = — /^i, 



Oo = 02 (z^, «"") = — /U.2- 



* In the present case the values of the a's giving the identical transformation 

 are a^^) = Oo'"' = 0. 



t Cf. Lie : Transformationsgruppen, III. G07 et seq. 



